P-CENTRAL GROUPS AND POINCARÉ DUALITY

Thomas S. Weigel

November 25^{rd}, 1996

§1 Introduction.

In general it is difficult to calculate the mod p cohomology ring of a finite group

H^{•}(G): = H^{•}(G, 𝔽_{p}) = Ext_{𝔽_{p}G}^{•}(𝔽_{p}, 𝔽_{p}) = H^{•}(BG, 𝔽_{p})

even for small examples. However, it is rather easy to determine H^{•}(G) for finite abelian groups (cf. [B_{1}, §3.5.]). The reason for this phenomenon is that for abelian groups, BG carries in addition the structure of an H-space, and hence H^{•}(G) has the structure of an 𝔽_{p}-Hopf algebra.

In this paper we consider finite groups G in which every element of order p is contained in the center of G. These groups will be called p-central. One can think of p-centrality as a generalization of p-torsionfreeness. For p-central groups G the cohomology ring H^{•}(G) carries the structure of a (left) A-comodule algebra for some 𝔽_{p}-Hopf algebra A. E.g., this fact can be used to show that H^{•}(G) is a Cohen-Macaulay algebra (cf. [Du], [BrH]) and hence, satisfies some weak form of Poincaré duality which was introduced by D.Benson and J.F.Carlson (cf. [BC]), In this case the Poincaré series

p(t): = p_{H^{•}(G)}(t) = ∑_{k∈ℕ_{0}} dim_{𝔽_{p}}(H^{k}(G)).t^{k}

satisfies the functional equation

p(1/t) = (−t)^{r}.p(t),  (1.1.)

where r: = log_{p}(|Ω_{1}(G)|).

Let K be a field. Classically, one says that a commutative^{1} connected ℕ_{0}-graded K-algebra A = ∐_{k∈ℕ_{0}} A_{k} satisfies Poincaré duality in dimension n, if

(a) dim_{K}(A_{k}) < ∞,

(b) dim_{K}(A_{n}) = 1 and dim_{K}(A_{k}) = 0 for k > n,

(c) the bilinear mapping A_{k} ⊗ A_{n−k} → A_{n} induced by the product is nondegenerated for k = 0, .., n.

Typeset by AMS-TEX

1This definition follows [MM]. Sometimes these algebras are also called anti-commutative.

In particular, this implies that p_{A}(t) satisfies the functional equation

p_{A}(1/t) = t^{−n}.p_{A}(t).  (1.2.)

The standard examples of Poincaré duality algebras in cohomology are the cohomology algebras of compact orientable manifolds (cf. [BoT, Chap.I, $5]), or the cohomology of unimodular finite dimensional K-Lie algebras H^{•}(L, K) (cf. [Ko, Th.7.1.]). Another example arises by considering the Galois cohomology of p-adic analytic groups. For these groups Poincaré duality of the Galois cohomology characterizes p-torsionfreeness, i.e., one has the following theorem (cf. [La, Chap.V, (2.5.)], [Qu_{1},§13], [Se, Chap.V]):

Theorem A. (M.Lazard, D.Quillen, J.-L.Verdier) Let Ĝ be a p-adic analytic group. Then the following is equivalent:

(1) H_{gal}^{•}(Ĝ, 𝔽_{p}) is of finite dimension, dim_{𝔽_{p}}(H_{gal}^{n}(Ĝ, 𝔽_{p})) ≠ 0, and for all k > n, dim_{𝔽_{p}}(H_{gal}^{k}(Ĝ, 𝔽_{p})) = 0,

(2) H_{gal}^{•}(Ĝ, 𝔽_{p}) satisfies Poincaré duality in dimension n.

(3) Ĝ is p-torsionfree and of dimension n.

Actually, the main subject of this paper will be p-central groups with a certain extension property (ΩEP) for p odd. The standard examples are p-groups P which have a faithful ℤ_{p}-linear representation ϕ: P → GL(W) on some finite ℤ_{p}-module W such that im(ϕ) ≤ id_{W} + p. End_{ℤ_{p}}(W). We will call these groups p-linear. The combinatorial properties of p-central groups have been studied in [W]. In analogy to the Theorem of D.Quillen, M.Lazard and J.-L.Verdier we will prove in this paper the following

Theorem B. Let G be a finite group and p odd. Then the following is equivalent:

(1) H^{•}(G)_{red} ≃ 𝔽_{p}[y_{1}, .., y_{n}], where y_{i} are homogeneous generators of degree 2,

(2) H^{•}(G) ≃ 𝔽_{p}[y_{1}, .., y_{n}]⊗C, where y_{i} are homogeneous generators of degree 2 and C is a finite-dimensional connected commutative 𝔽_{p}-algebra satisfying Poincaré duality in dimension n,

(3) G ≃ P ⋉ O_{p′}(G), where P ∈ Syl_{p}(G) is p-central with ΩEPand n = log_{p}(|Ω_{1}(P)|).

As an immediate consequence we can characterize all finite groups which mod p cohomology rings are isomorphic to the cohomology rings of abelian p-group.

Corollary C. Let G be a finite group, p odd. Then the following is equivalent:

(1) H^{•}(G) ≃ Λ[x_{1}, .., x_{n}] ⊗ 𝔽_{p}[y_{1}, .., y_{n}], where x_{i}, y_{j} are homogeneous generators with |x_{i}| = 1, |y_{j}| = 2.

(2) G ≃ P ⋉ O_{p′}(G), where P ∈ Syl_{p}(G) is powerful and p-central with ΩEP.

At this point it should be remarked that indeed every powerful p-central group with ΩEP is p-linear provided p ≥ 5 (cf. [W, Th.7.1.]).

§ 2. A SPLITTING THEOREM FOR COMODULE ALGEBRAS. 3

The proof of Theorem B requires various techniques. First we prove some splitting theorem for a certain type of comodule algebras. Together with a detailed analysis of the Lyndon-Hochschild-Serre spectral sequence associated to 2 types of group extensions, it is possible to determine the structure of H^{•}(P) for P a finite p-central group with ΩEP, p odd. Essential for this treatment is the knowledge of the differential d_{2}: E_{2}^{p,q} → E_{2}^{p+2,q−1} (cf. [CV]). But maybe also other techniques may yield this result (cf. Remark 7.1.).

The stratification theorem of D.Quillen (cf. [Qu_{1}]) allows to analyze the structure of a group G with given mod p cohomology ring. Furthermore, his cohomological p-nilpotency criterium (cf. [Qu_{2}]) can be used in this context very successfully. These two facts will show that (1) implies (3).

So far we have restricted our consideration to the case that p is odd. For p = 2 several problems arise and we were not able to deduce a theorem like Theorem B for p = 2. It might be the case that the right analogue for even p is Corollary 5.18.2 in [B_{2}].

§ 2. A splitting theorem for comodule algebras.

Throughout this section let K denote a field. For us the underlying vectorspaces of all K-algebras, modules, K-coalgebras, comodules,..etc will be ℕ_{0}-graded Kvectorspaces. We are following the treatment given in [MM]. A K-vectorspace A = ∐_{k∈ℕ_{0}} A_{k} is called connected if A_{0} ≃ K, and of finite type if dim_{K}(A_{k}) < ∞ for all k ∈ ℕ_{0}.

Let (A, Δ_{A}, ε_{A}) be a K-coalgebra. If (M, Γ_{M}) is a right A-comodule and (N, Δ_{N}) is a left A-comodule, one defines the cotensor product by M □_{A}N: = ker(γ), where

γ: M ⊗ N ⟶^{Γ_{M} ⊗id_{N} −id_{M} ⊗Δ_{M}} M ⊗ A ⊗ N.  (2.1.)

Note that this makes M □_{A}N an ℕ_{0}-graded K-module. Moreover, if M is an A-bicomodule, M □_{A}N possesses canonically the structure of a left A-comodule.

A K-Hopf algebra A = ∐_{k∈ℕ_{0}} A_{k} is a graded K-module together with morphisms of graded K-modules

ϕ_{A}:A ⊗ A ⟶ A,  η_{A}:K ⟶ A,
Δ_{A}:A ⟶ A ⊗ A,  ε_{A}:A ⟶ K

such that (1) (A, ϕ_{A}, η_{A}, ε_{A}) is an augmented K-algebra,

(2) (A, Δ_{A}, ε_{A}, η_{A}) is an augmented K-coalgebra,

(3) Δ_{A}: A → A ⊗ A is a morphism of K-algebras,

(4) ϕ_{A}: A ⊗ A → A is a morphism of K-coalgebras. Modules respectively comodules for Hopf-algebras, are modules respectively comodules for the underlying algebra respectively coalgebra structure of the Hopf algebra. Note that if A is a K-Hopf algebra, then K is canonically an A-bimodule and also an A-bicomodule.

Let (M, ϕ_{M}), (N, ϕ_{N}) be left A-modules for some K-Hopf algebra A. Then (M ⊗ N, ϕ_{M⊗N}) is a left A-module, where

ϕ_{M⊗N}: A ⊗ M ⊗ N ⟶^{Δ_{A}⊗id_{M} ⊗id_{N}} A ⊗ A ⊗ M ⊗ N ⟶^{id_{A}⊗T ⊗id_{N}} ....
A ⊗ A ⊗ M ⊗ A ⊗ N ⟶^{ϕ_{M} ⊗ϕ_{N}} M ⊗ N,  (2.2.)

where T: M ⊗ A → A ⊗ M denotes the twisting operator, i.e., for homogeneous elements m ∈ M_{p}, a ∈ A_{q} one has

T (m ⊗ a): = (−1)^{pq}a ⊗ m.  (2.3.)

Moreover, if (M, Δ_{M}), (N, Δ_{N}) are left A-comodules, then (M ⊗ N, Δ_{M⊗N}) is a left A-comodule, where

Δ_{M⊗N}: M ⊗ N ⟶^{Δ_{M} ⊗Δ_{N}} A ⊗ M ⊗ A ⊗ N ⟶^{id_{A}⊗T ⊗id_{N}} ....
A ⊗ A ⊗ M ⊗ N ⟶^{ϕ_{A}⊗id_{M} ⊗id_{N}} A ⊗ M ⊗ N.  (2.4.)

Let (A, ϕ_{A}, Δ_{A}, η_{A}, ε_{A}) be a K-Hopf algebra, B = ∐_{k∈ℕ_{0}} B_{k} a graded K-module and

ϕ_{B}:B ⊗ B ⟶ B,  η_{B}:K ⟶ B
Δ_{B}:B ⟶ A ⊗ B,  ε_{B}:B ⟶ K

be morphisms of graded K-modules such that (1) (B, Δ_{B}) is a left A-comodule,

(2) (B, ϕ_{A}, η_{B}, ε_{B}) is an augmented K-algebra,

(3) ϕ_{B}: B ⊗ B → B is a morphism of left A-comodules,

(4) η_{B}: K → B is a morphism of left A-comodules, Then (B, ϕ_{B}, Δ_{B}, η_{B}, ε_{B}) is called an augmented left A-comodule algebra. These objects will be of great interest for us, since the 𝔽_{p}-cohomology ring of a group with a finite p-central subgroup is an augmented left A-comodule algebra for some 𝔽_{p}-Hopf algebra A.

For an augmented left A-comodule algebra B one has a canonical mapping j: B → A given by

j: B ⟶^{Δ_{B}} A ⊗ B ⟶^{id_{A}⊗ε_{B}} A ⊗ K ⟶^{ϕ_{A}∘(id_{A}⊗η_{A})} A  (2.5.)

which is a morphism of graded K-modules. If one gives in contrast to (2.4.) A ⊗ B the left A-comodule structure defined by Δ_{A⊗B}: = Δ_{A} ⊗ id_{B}, one sees easily that all morphisms are morphisms of left A-comodules. Hence j: B → A is a morphism of left A-comodules.

Let C: = K □_{A}B denote the cotensor produkt of the A-bicomodule K with the left A-comodule B. Note that C is a trivial left A-subcomodule of K ⊗ B. Let ι_{C}: C → K ⊗ B denote the canonical embedding. Then one has a canonical mapping i: C → B given by

i: C ⟶^{ι_{C}} K ⊗ B ⟶^{η_{B}⊗id_{B}} B ⊗ B ⟶^{ϕ_{B}} B  (2.6.)

which is a morphism of left A-comodules. This morphism has the following property:

Proposition 2.1. If A is a K-Hopf algebra, B an augmented left A-comodule algebra and C: = K □_{A}B. Then there exist uniquely determined mappings of graded K-modules

ϕ_{C}: C ⊗ C ⟶ C, η_{C}: K ⟶ C

such that (C, ϕ_{C}, η_{C}) is a K-algebra and i: C → B is an injective morphism of Kalgebras. Furthermore, (C, ϕ_{C}, η_{C}, ε_{C}) with ε_{C}: = ε_{B} ∘i is an augmented K-algebra.

Proof. Since we assume that K is a field, i is certainly injective. The existence and uniqueness follows from Proposition 4.6. in [MM]. The last remark is an immediate consequence as i and ε_{B} are morphisms of K-algebras.

Let A be a K-Hopf algebra. We call A of affine type, if:

(1) as K-algebra A is isomorphic to a free commutative K-algebra,

A ≃ Λ[ x_{μ} | μ ∈ S_{1} ] ⊗ K[ x_{λ} | λ ∈ S_{2} ]

for some sets S_{1}, S_{2}, where for char(K) ≠ 2, x_{μ}, μ ∈ S_{1}, are homogeneous elements of odd degree, x_{λ}, λ ∈ S_{2}, are homogeneous elements of even degree, for char(K) = 2, S_{1} = ∅ and x_{λ}, λ ∈ S_{2}, are homogeneous elements of positive degree.

(2) For all λ ∈ S_{1} ∪ S_{2}, x_{λ} are primitive elements, i.e.,

x_{λ} ∈ P (A): = { a ∈ A | Δ_{A}(a) = 1_{A} ⊗ a + a ⊗ 1_{A} }.

Note that (1) and (2) determine the K-Hopf algebra structure (ϕ_{A}, Δ_{A}, η_{A}, ε_{A}) on A uniquely. We recall the following theorem from [MM]:

Theorem 2.2. Let A be a connected K-Hopf algebra, B a connected augmented A-comodule algebra, C = K □_{A}B, and assume that the canonical morphism j: B → A as given in (2.5.) is surjective. Then

(1) there exists a morphism g: A → B of left A-comodules such that j ∘g = id_{A}.

(2) for every morphism g: A → B of left A-comodules with j ∘ g = id_{A}, the composition

h: A ⊗ C ⟶^{g⊗i} B ⊗ B ⟶^{ϕ_{B}} B

is simultaneously an isomorphism of left A-comodules and right C-modules.

Proof. See Theorem 4.7. and the proof of Theorem 4.4. in [MM].

For K-Hopf algebras of affine type this theorem specializes to the following:

Theorem 2.3. Let A be a K-Hopf algebra of affine type and B a connected augmented left A-comodule algebra. Assume further that the canonical homomorphism of A-comodules j: B → A is surjective and additionally also a morphism of K-algebras. Then

(a) there exists a section σ: A → B which is an injective homomorphism of K-algebras and A-comodules satisfying j ∘ σ = id_{A}.

(b) Let C: = K □_{A}B. Then there exists an isomorphism

h^{∗} = ϕ_{B} ∘ (σ ⊗ i): A ⊗ C ⟶ B

which is simultaneously an isomorphism of left A-comodules, right C-mo-dules and left A-modules, where the left A-module structure on B and A⊗C is given by

ψ_{B} = ϕ_{B} ∘ (σ ⊗ id_{B}):A ⊗ B ⟶ B,
ψ_{A⊗C} = ϕ_{A} ⊗ id_{C}:  A ⊗ A ⊗ C ⟶ A ⊗ C.

(c) If B is a commutative K-algebra, i.e., T ∘ ϕ_{B} = ϕ_{B}, then h^{∗} as given in (b) is an isomorphism of K-algebras.

Proof. (a) Let

P_{j}(B): = { b ∈ B | Δ_{B}(b) = 1_{A} ⊗ b + j(b) ⊗ 1_{B} }

denote the j-primitive elements in the left A-comodule B. As j is a morphism of K-algebras, one has j(1_{B}) = 1_{A}. Since j is a morphism of left A-comodules, it follows that

j_{∗}: = j|_{P_{j}(B)}: P_{j}(B) ⟶ P (A).

We claim that j_{∗} is surjective. Let C: = K □_{A}B denote the cotensor product of the left A-comodule B and the A-bicomodule K. Let i: C → B denote the canonical injection which is a morphism of K-algebras (cf. (2.6.)). By Theorem 2.2.(1), there exists a morphism of A-comodules g: A → B satisfying j ∘ g = id_{A}. As A and B are connected, one obtains g(1_{A}) = 1_{B}. By Theorem 2.2.(2), the composition

h: A ⊗ C ⟶^{g⊗i} B ⊗ B ⟶^{ϕ_{B}} B,

is an isomorphism of left A-comodules. Hence h(A ⊗ 1_{C}) is a left A-subcomodule of B with

j ∘ h(a ⊗ 1_{C}) = a, for all a ∈ A.  (2.7.)

Let x ∈ P (A) be a primitive element in A. Hence it follows that

Δ_{B}(h(a ⊗ 1_{C})) = (id_{A} ⊗ h) ∘ Δ_{A⊗C}(a ⊗ 1_{C})
= (id_{A} ⊗ h)(a ⊗ 1_{A} ⊗ 1_{C} + 1_{A} ⊗ a ⊗ 1_{C})
= a ⊗ h(1_{A} ⊗ 1_{C}) + 1 ⊗ h(a ⊗ 1_{C})
= a ⊗ 1_{B} + 1_{A} ⊗ h(a ⊗ 1_{C})

By (2.7.), one concludes that h(a ⊗ 1) ∈ P_{j}(B). Hence, j_{∗} is surjective.

By hypothesis, A = Λ[x_{μ}|μ ∈ S_{1}] ⊗ K[x_{λ}|λ ∈ S_{2}], where all x_{λ}, λ ∈ S_{1} ∪ S_{2}, are primitive homogeneous elements of positive degree. Let t_{λ} ∈ P_{j}(B) be a homogeneous preimage of x_{λ} under j, i.e., j(t_{λ}) = x_{λ} for all λ ∈ S_{1} ∪ S_{2}. Let D: = ⟨t_{λ}|λ ∈ S_{1} ∪S_{2}⟩_{K−alg} ≤ B denote the K-subalgebra of B generated by the elements t_{λ} and σ: A → D denote the induced homomorphism of graded K-algebras which exists, since A is a free commutative K-algebra. Then by construction σ∘j|_{D} = id_{D} and j|_{D}∘σ = id_{A}. Hence σ is an isomorphism of graded K-algebras. Thus it remains to prove that D is a left A-subcomodule of B.

By induction, it suffices to prove, that for t′ ∈ D with Δ_{B}(t′) ∈ A ⊗ D and t_{λ} ∈ D given as above, one gets

Δ_{B}(t′.t_{λ}) ∈ A ⊗ D,

where we have put t′.t_{λ}: = ϕ_{B}(t′ ⊗ t_{λ}). As B is a left A-comodule algebra, one has

Δ_{B}(t′.t_{k}) = Δ_{B}(ϕ_{B}(t′ ⊗ t_{K}))
= (ϕ_{A} ⊗ ϕ_{B}) ∘ (id_{A} ⊗ T ⊗ id_{B})(Δ_{B}(t′) ⊗ Δ(t_{k})),

where T: B ⊗ A → A ⊗ B denotes the twisting operator (cf. (2.3.)). Since Δ_{B}(t′), Δ_{B}(t_{k}) ∈ A ⊗ D one concludes that Δ_{B}(t′) ⊗ Δ_{B}(t_{λ}) ∈ A ⊗ D ⊗ A ⊗ D. As D ≤ B is a K-subalgebra, Δ_{B}(t′.t_{λ}) ∈ A ⊗ D and this completes the proof of the first part of the theorem.

(b) By the proof of Theorem 4.7. in [MM], one knows that for any morphism g: A → B of left A-comodules with j ∘ g = id_{A}, the composition

h: A ⊗ C ⟶^{g⊗i} B ⊗ B ⟶^{ϕ_{B}} B

is an isomorphism of left A-comodules and right C-modules. Let

h^{∗}: A ⊗ C ⟶^{σ⊗i} B ⊗ B ⟶^{ϕ_{B}} B

with σ: A → B as in part (a). Hence h^{∗} is an isomorphism of left A-comodules and right C-modules. Furthermore, for a′, a ∈ A and c ∈ C one obtains

h^{∗} ∘ ψ_{A⊗C}(a′ ⊗ a ⊗ c) = h^{∗}(a′.a ⊗ c)
= σ(a′).σ(a).i(c) = ψ_{B}(a′ ⊗ h^{∗}(a ⊗ c)).

Hence h^{∗} is also an isomorphism of left A-modules.

(c) Since h^{∗}(1_{A} ⊗ 1_{C}) = 1_{B}, it suffices to show that the diagram

A ⊗ C ⊗ A ⊗ C ⟶^{ϕ_{A⊗C}} A ⊗ C
h^{∗}⊗h^{∗}↓  ↓_{h^{∗}}
B ⊗ B  ⟶^{ϕ_{B}} B

commutes. Let a_{1}, a_{2} ∈ A, c_{1}, c_{2} ∈ C be homogeneous elements with a_{2} of degree p and c_{1} of degree q. Hence one obtains

ϕ_{B} ∘ (h^{∗} ⊗ h^{∗})(a_{1} ⊗ c_{1} ⊗ a_{2} ⊗ c_{2}) = σ(a_{1}).i(c_{1}).σ(a_{2}).i(c_{2})

and

h^{∗} ∘ ϕ_{A⊗C}(a_{1} ⊗ c_{1} ⊗ a_{2} ⊗ c_{2}) = (−1)^{pq}.σ(a_{1}).σ(a_{2}).i(c_{1}).i(c_{2}).

Hence the commutativity of the considered diagram follows from the commutativity of the K-algebra B and this completes the proof of the theorem.

If the canonical morphism j: B → A is not surjective, one still has the following property, which has been used already in [BrH] to analyze the structure of 𝔽_{p}- cohomology rings:

Proposition 2.4. Let A be a connected cocommutative K-Hopf algebra and B a connected augmented left A-comodule algebra. Assume further that j: B → A is a morphism of K-algebras. Then j is a morphism of augmented left A-comodule algebra. In particular, j(B) ≤ A is a sub-Hopf algebra of A.

Proof. The map j is a morphism of A-comodules (cf. (2.5.)) and by hypothesis also of K-algebras. Hence, it is a morphism of augmented left A-comodule algebras. Thus D: = j(B) ≤ A is a connected augmented left A-comodule algebra. In particular, Δ_{A}(D) ≤ A ⊗ D. Since A is cocommutative, one has also Δ_{A}(D) ≤ D ⊗ A. Hence, Δ_{A} ≤ D ⊗ D and the claim follows.

§ 3. The cohomology of groups with p-central subgroups.

Let H^{•}( ) denote the mapping assigning every group G its graded 𝔽_{p}-cohomology ring H^{•}(G): = H^{•}(G, 𝔽_{p}). Every morphism of groups ϕ: G → H induces canonically a morphism

ϕ^{•}: = H^{•}(ϕ): H^{•}(H) → H^{•}(G)

of connected augmented commutative ℕ_{0}-graded 𝔽_{p}-algebras. Hence

H^{•}( ): Gr ⟶ 𝔽_{p}-alg^{•}

is a contravariant functor from the category Gr of groups to the category 𝔽_{p}-alg^{•} of connected augmented commutative ℕ_{0}-graded 𝔽_{p}-algebras.

Furthermore, H^{•}( ) is a functor of monoidal categories, i.e., for all G, H ∈ obj(Gr) one has a canonical isomorphism of 𝔽_{p}-algebras

H^{•}(G) ⊗ H^{•}(H) ≃ H^{•}(G × H),

and the twisting isomorphism t_{G×H}: G × H → H × G induces the twisting operator H^{•}(t_{G×H}) = T as given in (2.3). Hence, the 𝔽_{p}-algebra structure on H^{•}(G) is given by the mappings ϕ_{G}^{•}, η_{G}^{•}, ε_{G}^{•}, where ϕ_{G}: G → G × G is the diagonal embedding, and η_{G}: G → 1, ε_{G}: 1 → G are the trivial mappings.

If G is abelian, there are additionally canonical morphisms of groups Δ_{G}: G × G → G and S_{G}: G → G given by Δ_{G}(g, h): = g.h and S_{G}(g): = g^{−1}, making (H^{•}(G), ϕ_{G}^{•}, η_{G}^{•}, Δ_{G}^{•}, ε_{G}^{•}, S_{G}^{•}) a commutative, cocommutative 𝔽_{p}-Hopf algebra with antipode.

If G is a finite group, H^{•}(G) is of finite type. Hence, as 𝔽_{p} is a field, there is a canonical isomorphism

Hom(H^{•}(G, 𝔽_{p}), 𝔽_{p}) ⟶ H_{•}(G, 𝔽_{p}).

With this identification, the by Δ_{G}^{•} induced mapping

(Δ_{G}^{•})^{∗}: H_{•}(G, 𝔽_{p}) ⊗ H_{•}(G, 𝔽_{p}) ⟶ H_{•}(G, 𝔽_{p})

just coincides with the Pontryagin-multiplication in homology (cf. [McC, §4.4.]).

§ 3. THE COHOMOLOGY OF GROUPS WITH P-CENTRAL SUBGROUPS. 9

Let p be a prime and Ω_{k}( ): obj(Gr) → obj(Gr) be given by

Ω_{k}(H): = ⟨ h ∈ H | h^{p^{k}} = 1 ⟩  (3.1.)

for all H ∈ obj(Gr). This definition actually depends on the prime p. However, since it will be always clear from the context for which prime p we consider the group Ω_{k}(G), we omit it in the notation.

Let G be a group and Ω ≤ G a subgroup. Then Ω is called a p^{k}-central subgroup provided

Ω ≤ Ω_{k}(Z(G)).  (3.2.)

If Ω_{k}(G) ≤ Ω_{k}(Z(G)), we call the group G itself p^{k}-central.

From now on let G denote a group and Ω a finite p-central subgroup for p odd, respectively a finite 4-central subgroup isomorphic to a homocyclic group of exponent 4 for p = 2, i.e.,

Ω ≤ Ω_{1}(Z(G)),  p odd,
Ω ≤ Ω_{2}(Z(G)), Ω ≃ (ℤ/4.ℤ)^{r}, r ∈ ℕ, p = 2, (3.3.)

Recall that a group G is called homocyclic, if it isomorphic to a finite direct product of cyclic groups of the same order.

Let j: Ω → G denote the canonical embedding. In this case one has a canonical mapping Δ_{G}: Ω × G ⟶ G given by

Δ_{G}: = m_{G} ∘ (j × id_{G}),

where m_{G}: G × G → G denotes multiplication in G. Since im(j) ≤ Z(G), Δ_{G} is a morphism of groups. Hence, one obtains a morphism of 𝔽_{p}-algebras

Δ_{G}^{•}: H^{•}(G) ⟶ H^{•}(Ω) ⊗ H^{•}(G).

As the diagram

Ω × Ω × G ⟶^{id_{Ω}×Δ_{G}} Ω × G
Δ_{Ω}×id_{G}↓  ↓^{Δ_{G}}
Ω × G ⟶^{Δ_{G}} G

commutes, it is easily seen that Δ_{G}^{•} defines a left H^{•}(Ω)-comodule structure on H^{•}(G). Furthermore, the diagram

\begin{matrix}G × G & ⟵^{ϕ_{G}} & G \\ ↑^{δ} & & ↑^{Δ_{G}} \\ Ω × G × G & ⟵^{id_{Ω}×ϕ_{G}} & Ω × G \end{matrix}  (3.4.)

commutes, where δ is given by δ(ω, g_{1}, g_{2}): = (ω.g_{1}, ω.g_{2}), ω ∈ Ω, g_{1}, g_{2} ∈ G. Hence, δ is the composition of the mappings

δ: Ω × G × G ⟶^{ϕ_{Ω}×id_{G}×id_{G}} Ω × Ω × G × G ⟶^{id_{Ω}×t_{Ω×G}×id_{G}} ....
Ω × G × Ω × G ⟶^{Δ_{G}×Δ_{G}} G × G.  (3.5.)

Thus

δ^{•}: H^{•}(G) ⊗ H^{•}(G) → H^{•}(Ω) ⊗ H^{•}(G) ⊗ H^{•}(G)

coincides with the canonical left H^{•}(Ω)-comodule structure given on the tensor product H^{•}(G)⊗H^{•}(G). From the commutativity of the diagram (3.4.), one obtains that

Δ_{G}^{•} ∘ ϕ_{G}^{•} = (id_{Ω}^{•} ⊗ ϕ_{G}^{•}) ∘ δ^{•},

i.e., ϕ_{G}^{•} is a morphism of left H^{•}(Ω)-comodules. With a similar argument one can show that η_{G}^{•} is also a morphism of left H^{•}(Ω)-comodules. But note that there is no reason why ε_{G}^{•} should be a morphism of left H^{•}(Ω)-comodules.

Consider the mapping j^{∘}: H^{•}(G) → H^{•}(Ω) given by

j^{∘} = ϕ_{Ω}^{•} ∘ (id_{Ω}^{•} ⊗ η_{G}^{•}) ∘ (id_{Ω}^{•} ⊗ ε_{G}^{•}) ∘ Δ_{G}^{•}  (3.6.)

(cf. (2.5)). Since

Ω × Ω ⟶^{id_{Ω}⊗(ε_{G}∘η_{Ω})} Ω × G
ϕ_{Ω}↑  ↓^{Δ_{G}}
Ω  ⟶^{j}  G

is a commutative diagram, one has j^{•} = j^{∘}, where j^{•} is the mapping induced from the canonical embedding j: Ω → G. Hence one obtains:

Fact 3.1. Let G be a group and Ω ≤ G a finite central subgroup as described in (3.3.) Then (H^{•}(G), ϕ_{G}^{•}, Δ_{G}^{•}, η_{G}^{•}, ε_{G}^{•}) is an augmented connected left H^{•}(Ω)- comodule algebra. Furthermore, the canonical mapping j^{•}: H^{•}(G) → H^{•}(Ω) is simultaneously a morphism of 𝔽_{p}-algebras and H^{•}(Ω)-comodules.

For A ∈ obj(𝔽_{p}-alg^{•}) we put

nil(A): = { a ∈ A | ∃k ∈ ℕ, a^{k} = 0 }.

Since A is commutative, nil(A) is a graded ideal in A. Hence

A_{red}: = A/nil(A) ∈ obj(𝔽_{p}-alg^{•})

is a reduced 𝔽_{p}-algebra, i.e. nil(A_{red}) = 0. By

ρ_{A}: A ⟶ A/nil(A)  (3.7.)

we will denote the canonical surjective mapping of 𝔽_{p}-algebras. Every morphism of commutative 𝔽_{p}-algebras ψ: A → B induces a morphism of reduced strictly commutative 𝔽_{p}-algebras ψ_{red}: A_{red} ⟶ B_{red} given by

ψ_{red}(a + nil(A)): = ψ(a) + nil(B).  (3.8.)

Note that ψ_{red} is characterized as the unique morphism of 𝔽_{p}-algebras satisfying

ρ_{B} ∘ ψ = ψ_{red} ∘ ρ_{A}.  (3.9.)

From this one deduces that _{red}: 𝔽_{p}-alg^{•} → 𝔽_{p}-alg^{•} is a covariant functor.

Let A, B ∈ obj(𝔽_{p}-alg^{•}). Considering A ⊗ B as a bi-graded 𝔽_{p}-algebra, one can show easily that (A ⊗ B)_{red} is canonically isomorphic to A_{red} ⊗ B_{red}, i.e.,

nil(A ⊗ B) = nil(A) ⊗ B + A ⊗ nil(B)  (3.10.)

Furthermore, if φ: A → B, ψ: C → D ∈ mor(𝔽_{p}-alg^{•}), then

(φ ⊗ ψ)_{red} = φ_{red} ⊗ ψ_{red}.  (3.11.)

Thus _{red} is a functor of monoidal categories. Furthermore, if A is a 𝔽_{p}-Hopf algebra, one concludes from (3.8.) and (3.10.), that nil(A) is a coideal. Hence, A_{red} has a canonical 𝔽_{p}-Hopf algebra structure.

If H is a cyclic group of order p, p odd, respectively a cyclic group of order 4, p = 2, one knows that H^{•}(H)_{red} ≃ 𝔽_{p}[x] is a 𝔽_{p}-Hopf algebra of affine type generated in degree 2. Since H^{•}( ) and _{red} are functors of monoidal categories, one follows that if H is an elementary abelian p-group, p odd, respectively a homocyclic 2-group of exponent 4, p = 2, one has

H^{•}(H)_{red} ≃ 𝔽_{p}[x_{1}, .., x_{r}] = 𝔽_{p}[x_{1}] ⊗ ⋯ ⊗ 𝔽_{p}[x_{r}],

where x_{j} are primitive homogeneous elements of degree 2, r denotes the p-rank of H and the isomorphism is an isomorphism of 𝔽_{p}-Hopf algebras with antipode.

Let G be a group and Ω ≤ Z(G) be a finite central subgroup as described in (3.3.). For simplicity we put Â: = H^{•}(Ω),

(A, ϕ_{A}, η_{A}, Δ_{A}, ε_{A}): = (H^{•}(Ω)_{red}, ϕ_{Ω,red}^{•}, η_{Ω,red}^{•}, Δ_{Ω,red}^{•}, ε_{Ω,red}^{•})
and  (B, ϕ_{B}, η_{B}, ε_{B}): = (H^{•}(G), ϕ_{G}^{•}, η_{G}^{•}, ε_{G}^{•}). (3.12.)

A is a 𝔽_{p}-Hopf algebra of affine type, and B is a connected augmented 𝔽_{p}-algebra. We put Δ_{B}: = (ρ ⊗ id_{B}) ∘ Δ_{G}^{•}, where ρ: Â → A denotes the canonical projection. Hence, Δ_{B} defines a left A-comodule structure on B. By (3.4.), the lefthand side of the diagram

B ⊗ B ⟶^{δ} Â ⊗ B ⊗ B ⟶^{ρ⊗id_{B⊗B}} A ⊗ B ⊗ B
ϕ_{B} ↓  ↓^{idÂ⊗ϕ_{B}}  ↓^{id_{A}⊗ϕ_{B}}
B ⟶^{Δ_{G}^{•}}  Â ⊗ B  ⟶^{ρ⊗id_{B}}  A ⊗ B

commutes, where δ is given as in (3.5.). But is easy to check that also the righthand side of the diagram commutes. This shows that ϕ_{B} is a morphism of left A-comodules. With a similar argument one shows that η_{B}: 𝔽_{p} → B is a morphism of left A-comodules.

The canonical mapping of left A-comodules is given by

j_{▲}: B ⟶^{Δ_{B}} A ⊗ B ⟶^{id_{A}⊗ε_{B}} A ⊗ 𝔽_{p} ⟶^{ϕ_{A}∘(id_{A}⊗η_{A})} A

(cf. (2.5.)). Furthermore, it follows straight from the definition of Δ_{B} and the 𝔽_{p}-Hopf algebra structure on A, that

j_{▲} = ρ ∘ j^{∘},  (3.13.)

where j^{∘} is defined as in (3.6.). Thus we obtain:

Fact 3.2. Let G be a group and Ω ≤ G a finite central subgroup of G as described in (3.3.). Then H^{•}(G) is a connected augmented left H^{•}(Ω)_{red}-comodule algebra. Furthermore, the canonical mapping

j_{▲}: H^{•}(G) ⟶ H^{•}(Ω)_{red}

coincides with the reduced restriction mapping ρ ∘ j^{∘} = ρ ∘ j^{•}, and thus is also a morphism of 𝔽_{p}-algebras.

After all these introducing remarks, we can know apply Theorem 2.3., to obtain the following ‘Splitting Theorem’:

Theorem 3.3. Let G be a group and let Ω ≤ G be a finite p-central subgroup if p is odd, or a homocyclic 4-central subgroup of G if p = 2. Assume further that the reduced restriction mapping

j_{▲} = ρ ∘ j^{•}: H^{•}(G) ⟶ H^{•}(Ω)_{red}

is surjective. Then

H^{•}(G) ≃ H^{•}(Ω)_{red} ⊗ C

and the isomorphism is an isomorphism of 𝔽_{p}-algebras and left H^{•}(Ω)_{red}-comodu-les, where

C: = 𝔽_{p} □_{H^{•}(Ω)_{red}}H^{•}(G)

denotes the cotensor product (cf. (2.1.) and Proposition 2.1.).

Proof. By Fact 3.2., one knows that that B: = H^{•}(G) is a connected augmented left A-comodule algebra, where A: = H^{•}(Ω)_{red} is of affine type. Hence the desired result follows from Theorem 2.3.(b) and (c).

Theorem 3.3. can be applied to describe the structure of the mod p cohomology ring of finite p-central groups for which j_{▲}: H^{•}(G) → H^{•}(Ω_{1}(G))_{red} is surjective. For p = 2, one actually has to consider a stronger condition than 2-centrality.

Definition 3.4. Let p = 2 and G a finite 4-central group. We call G 4^{∗}-central if Ω_{2}(G) is a homocylic group of exponent 4.

We simplify our notation and put

Ω(G): = Ω_{1}(G)  for p odd,
Ω(G): = Ω_{2}(G)  for p = 2. (3.14.)

Furthermore, for G a finite p-central group, let

l-rk_{p}(G): = dim_{𝔽_{p}}(Ω_{1}(G)).  (3.15.)

Then Theorem 3.3. implies:

§ 4. P -CENTRAL GROUPS WITH THE Ω-EXTENSION PROPERTY. 13

Corollary 3.5. Let G be a finite p-central group for p odd, respectively a finite 4^{∗}-central group for p = 2. Assume that the reduced restriction mapping

j_{▲}: = ρ ∘ j^{•}: H^{•}(G) ⟶ H^{•}(Ω(G))_{red}

is surjective. Then

H^{•}(G) ≃ C ⊗ 𝔽_{p}[y_{1}, .., y_{r}],

where the isomorphism is an isomorphism of 𝔽_{p}-algebras, C is a finite-dimensional 𝔽_{p}-algebra, r = l-rk_{p}(G), and all generators y_{i} are of degree 2.

Proof. By a theorem of B.B.Venkov and L.Evens, H^{•}(G) is a finitely generated 𝔽_{p}-algebra (cf. [Ev, Cor. 7.4.6]). Furthermore, it follows from Quillen’s stratification theorem (cf. [Qu_{1}]), that ker(j_{▲}) consists of nilpotent elements. Hence, C: = 𝔽_{p}□_{H^{•}(Ω(G))_{red}}H^{•}(G) is a finitely generated 𝔽_{p}-algebra consisting of nilpotent elements, and thus is finite.

We close this section with a remark concerning the general structure of the mod p cohomology ring of a finite p-central group G. In this case, it is known that the reduced restriction mapping determines H^{•}(G)_{red} up to isomorphism (cf. [BrH]).

Proposition 3.6. Let G be a finite p-central group. Then the reduced restriction mapping

j_{red}^{•}: H^{•}(G)_{red} ⟶ H^{•}(Ω_{1}(G))_{red}

is an injective inseparable isogeny. In particular,

H^{•}(G)_{red} ≃ 𝔽_{p}[y_{1}, .., y_{r}],

where r = l-rk_{p}(G) and y_{i} is of degree 2.p^{α_{i}}, α_{i} ∈ ℕ_{0} for p odd, respectively of degree 2^{α_{i}}, α_{i} ∈ ℕ_{0}, for p = 2.

Proof. By Quillen’s stratification theorem, j_{red}^{•} must be an injective inseperable isogeny. Furthermore, by Proposition 2.4, im(j_{red}^{•}) is a sub-Hopf algebra of H^{•}(Ω_{1}(G))_{red} and this yields the desired result.

§ 4. P -central groups with the Ω-extension property.

In this section we investigate finite p-central groups G with a property which is related to their mod p cohomology ring H^{•}(G). It is well-known that p-central groups, p odd, respectively 4-central groups are p-nilpotent (cf. [Hu, Kap.IV, Th.5.5]). This is the main reason why we restrict our considerations to finite pcentral p-groups. We will use the notations of the previous section.

Definition 4.1. Let G be a finite p-central p-group, p-odd, respectively a finite 4^{∗}-central 2-group. We say that G has the Ω-extension property (ΩEP), if there exists a finite p-central p-group Ḡ such that G ≃ Ḡ/Ω_{1}(Ḡ).

The combinatorial properties of p-central groups with ΩEP have been studies in [W, §5]. The connection of the ΩEP to the mod p cohomology ring is given by the following:

Theorem 4.2. Let G be a finite p-central p-group, p odd, or a finite 4^{∗}-central 2-group. Then the following is equivalent:

(1) H^{•}(G)_{red} is a polynomial 𝔽_{p}-algebra of Krull dimension l-rk_{p}(G) generated in degree 2.

(2) The reduced restriction mapping j_{▲}: H^{•}(G) → H^{•}(Ω(G))_{red} is surjective (cf. (4.1.)).

(3) The reduced restriction mapping j_{▲} is surjective in degree 2.

(4) G has the ΩEP.

Proof. Since G is p-central, j_{red}^{•}: H^{•}(G)_{red} → H^{•}(Ω(G))_{red} is injective (cf. Proposition 3.6.). Hence, if j_{▲} is surjective, j_{red}^{•} is an isomorphism. If H^{•}(G)_{red} is a polynomial 𝔽_{p}-algebra generated in degree 2, j_{red}^{•} must be an isomorphism as H^{•}(G)_{red} and H^{•}(Ω(G)) have the same Krull dimension. This shows that (1) and (2) are equivalent. Since j_{▲} is a morphism of ℕ_{0}-graded 𝔽_{p}-algebras, and as H^{•}(Ω(G))_{red} is generated by homogeneous elements of degree 2, (2) is certainly equivalent to (3). Hence, it suffices to show that (3) and (4) are equivalent.

Let S be a group and

c ∈ C^{2}(S, M) = { f ∈ 𝔉(S^{2}, M) | f(y, z)+f(x, yz) = f(xy, z)+f(x, y), x, y, z ∈ S }

denote a 2-cocycle with values in the trivial 𝔽_{p}S-module M. By S_{c}: = S × M we denote the group with multiplication

(s_{1}, m_{1}).(s_{2}, m_{2}): = (s_{1}.s_{2}, m_{1} + m_{2} + c(s_{1}, s_{2})).  (4.1.)

Let ι: M → S_{c}, π: S_{c} → M denote the canonical morphisms, i.e., one has

\begin{aligned}ι(m) & : = (1, m − c(1, 1)), & m ∈ M \\ π((s, m)) & : = s, & (s, m) ∈ S_{c} \end{aligned}  (4.2.)

Now let S be an elementary abelian p-group, p odd, respectively a homocyclic 2group of exponent 4, and c ∈ C^{2}(S, M), M a trivial 𝔽_{p}S-module. Then S_{c} is a p-group with nilpotency class less or equal to 2 satisfying γ_{2}(S_{c})^{p} = 1. Hence the mapping τ: S_{c} → M given by

τ(ω) = \{ \begin{matrix}ω^{p}, & p odd, \\ ω^{4}, & p = 2, \end{matrix}  ω ∈ S_{c}  (4.3.)

is a homomorphism of groups. This follows immediately from the Hall-Petrescu identity (cf. [HB, Chap.VIII, Lemma 1.1.]) or the fact that S_{c} is a powerful pgroup. Furthermore, ι(M) ≤ ker(τ). Hence τ induces a morphism τ_{c}^{∗}: S → M ∈ Hom(S, M) given by

τ_{c}^{∗}(s): = \{ \begin{matrix}c(1, 1) + ∑_{k=1}^{p−1} c(s^{k}, s), & p odd, \\ c(1, 1) + ∑_{k=1}^{3} c(s^{k}, s), & p = 2. \end{matrix}  (4.4.)

Note that for p′: = p − 1, p odd, respectively 2′: = 3, one has also

∑_{k=1}^{p′} c(s^{k}, s) = ∑_{k=1}^{p′} c(s, s^{k})  (4.5.)

Let χ: C^{2}(S, M) → Hom(S, M) denote the mapping given by

χ(c)(s): = τ_{c}^{∗}(s), c ∈ C^{2}(S, M), s ∈ S.  (4.6.)

Hence, χ is certainly surjective and 𝔽_{p}-linear. Let

f̂ ∈ B^{2}(S, M) = { f̂ ∈ 𝔉(S^{2}, M) | ∃f ∈ 𝔉(S, M) : f̂(x, y) = f(x) + f(y) − f(xy) }

be a coboundary. Then it follows by an elementary calculation that χ(f̂) = 0. Hence χ induces a mapping

χ^{∘}: H^{2}(S, M) → Hom(S, M)

given by χ^{∘}(c̄) = χ(c), where ¯ : C^{2}(S, M) → H^{2}(S, M) denotes the canonical projection.

Furthermore, if c̄ is a nilpotent element in the 𝔽_{p}-algebra H^{•}(S, 𝔽_{p}), then it can be represented by a cocycle c ∈ C^{2}(S, 𝔽_{p}) satisfying

c(x, y) = −c(y, x), c(x, x) = 0, for all x, y ∈ S.  (4.7.)

It follows from (4.5.) respectively the cocycle identity for p = 2 that χ^{∘}(c̄) = 0. Thus χ^{∘} induces a mapping

χ_{▲}: H^{2}(S, 𝔽_{p})_{red} ⟶ Hom(S, 𝔽_{p})  (4.8.)

given by χ_{▲}(ρ(c̄)) = χ^{∘}(c̄), where ρ: H^{2}(S, 𝔽_{p}) → H^{2}(S, 𝔽_{p})_{red} denotes the canonical projection (cf. (3.7.)). In this manner one obtains:

Lemma 4.3. Let χ_{▲}: H^{2}(S, 𝔽_{p})_{red} ⟶ Hom(S, 𝔽_{p}) be given as in (4.8.). Then χ_{▲} is a bijective linear mapping.

Proof. By construction χ_{▲} is certainly 𝔽_{p}-linear and surjective. Since

dim_{𝔽_{p}}(H^{2}(S, 𝔽_{p})_{red}) = dim_{𝔽_{p}}(Hom(S, 𝔽_{p})),

the desired result follows.

Let r: = rk_{p}(S), c_{1}, .., c_{r} ∈ C^{2}(S, 𝔽_{p}), and c: = (c_{1}, .., c_{r}) ∈ C^{2}(S, 𝔽_{p}^{r}) denote the corresponding cocyle with values in the trivial 𝔽_{p}S-module 𝔽_{p}^{r}. Let

1 ⟶ 𝔽_{p}^{r} ⟶^{ι} S_{c} ⟶^{π} S ⟶ 1  (4.9.)

denote the corresponding extension (cf. (4.1.)). Then for ω ∈ S_{c}, one gets

τ(ω) = ι(χ(c_{1})(π(ω)), .., χ(c_{r})(π(ω)))  (4.10.)

Hence one obtains the following:

Lemma 4.4. The following is equivalent:

(1) The extension (4.10.) satisfies τ(S_{c}) = ι(𝔽_{p}^{r}) = Ω_{1}(S_{c}).

(2) χ(c_{1}), .., χ(c_{r}) is a basis of Hom(S, 𝔽_{p}).

(3) χ_{▲}(ρ(c̄_{1})), .., χ_{▲}(ρ(c̄_{r})) is a basis of Hom(S, 𝔽_{p}).

Proof. It is obvious that (2) and (3) are equivalent. Assume that (1) holds. As τ(S_{c}) = ι(𝔽_{p}^{r}), one concludes from (4.11.) that χ(c_{1}), .., χ(c_{r}) span Hom(S, 𝔽_{p}). Hence, they have to be linearly independent. From (2) one concludes using (4.11) that τ(S_{c}) = ι(𝔽_{p}^{r}). If p is odd, one obtains by the usual order argument that

Ω_{1}(S_{c}) = ker(τ) = ι(𝔽_{p}^{r}).

For p = 2, it is straightforward that ι(𝔽_{2}^{r}) ≤ Ω_{1}(S_{c}). Equality follows from the fact that S_{c} is a powerful 2-group of rank r.

Now we can complete the proof of Theorem 4.2. Assume that G has the ΩEPand let Ḡ be a finite p-central group such that G = Ḡ/Ω_{1}(Ḡ). By [W, Remark 5.3], we may assume that l-rk(G) = l-rk(Ḡ) =: r. Thus Ḡ = G_{c}, with c: = (c_{1}, .., c_{r}) and

1 ⟶ 𝔽_{p}^{r} ⟶^{ι} G_{c} ⟶^{π} G ⟶ 1  (4.11.)

denotes the associated central extension. Let Ω̄(G): = π^{−1}(Ω(G)) (cf. (3.14.)). Since Ω_{1}(G_{c}) = ker(π), the induced extension

1 ⟶ 𝔽_{p}^{r} ⟶^{ι} Ω̄(G) ⟶^{π|_{Ω̄(G)}} Ω(G) ⟶ 1  (4.12.)

satisfies ι(𝔽_{p}) = Ω_{1}(Ω̄(G)). Since Ω̄(G) is powerful with Ω̄(G)^{p} = ι(𝔽_{p}^{r}) for p odd, respectively Ω̄(G)^{4} = ι(𝔽_{2}^{r}) for p = 2, one obtains τ(Ω̄(G)) = ι(𝔽_{p}^{r}) = Ω_{1}(Ω̄(G)).

Moreover, Ω̄(G) = Ω(G)_{R(c)} (cf. (4.1.)), where R(c) = (R(c_{1}), .., R(c_{r})) denotes the image of c under the restriction mapping R: C^{2}(G, 𝔽_{p}^{r}) → C^{2}(Ω(G), 𝔽_{p}^{r}). Since R̄(d) = j^{•}(d̄) for all d ∈ C^{2}(G, 𝔽_{p}), where j^{•}: H^{2}(G, 𝔽_{p}) → H^{2}(Ω(G), 𝔽_{p}) denotes the restriction mapping in cohomology, one gets from Lemma 4.4., that this is equivalent to the fact that χ_{▲}(j_{▲}(c̄_{1})), .., χ_{▲}(j_{▲}(c̄_{r})) is a basis of Hom(Ω(G), 𝔽_{p}). Since χ_{▲} is an isomorphism of 𝔽_{p}-spaces, this is equivalent that j_{▲} is surjective in degree 2.

Theorem 4.2 yields the following property of p-central groups with ΩEP.

Proposition 4.5. Let p be odd, and let G be a finite p-central p-group with ΩEP. Let A ≤ Ω_{1}(G) such that A ∩ Ω_{2}(G)^{p} = 1. Then Ḡ: = G/A is a p-central p-group with ΩEP.

Proof. By [W, Prop.4.1.(d)], Ḡ is a p-central p-group and satisfies l-rk_{p}(G) = l-rk_{p}(Ḡ) + l-rk_{p}(A). Consider the Lyndon-Hochschild-Serre spectral sequence associated to the extension

1 ⟶ A ⟶ G ⟶ Ḡ ⟶ 1.

§5. F -ISOMORPHISMS IN COHOMOLOGY. 17

By Theorem 4.2., Tot(E_{∞})_{red} is a polynomial 𝔽_{p}-algebra of Krull dimension r = l-rk_{p}(G) generated in degree 2. H^{•}(A) is a polynomial 𝔽_{p}-algebra of Krull dimension l-rk_{p}(A) generated in degree 2. Since Tot(E_{∞})_{red} is a subquotient of Tot(E_{2})_{red}, it follows that Tot(E_{2})_{red} has to contain a polynomial 𝔽_{p}-subalgebra of Krull dimension r generated in degree 2. But Tot(E_{2}) ≃ H^{•}(Ḡ) ⊗ H^{•}(A) and therefore Tot(E_{2})_{red} ≃ H^{•}(Ḡ)_{red} ⊗ H^{•}(A)_{red} (cf. (3.10.)). Since Ḡ is p-central H^{•}(Ḡ)_{red} is a polynomial 𝔽_{p}-algebra of Krull dimension l-rk_{p}(Ḡ) generated in degree 2.p^{α_{i}}, α_{i} ∈ ℕ_{0}, i = 1, .., l-rk_{p}(Ḡ) (cf. Prop.3.6.). Hence, H^{•}(Ḡ)_{red} must be generated in degree 2, and the claim follows from Theorem 4.2.

We close this section with the discussion of the basic examples.

Examples 4.6. If W is a finite ℤ_{p}-module, it has been shown in [W,§6] that for p odd, G = id_{W} + p. End_{ℤ_{p}}(W) is a powerful p-central group with ΩEP. We call a finite p-central p-group G p-linear, if there exists a finite ℤ_{p}-module W and a faithful ℤ_{p}-linear representation ϕ: G → GL(W) such that im(ϕ) ≤ id_{W} + p. End_{ℤ_{p}}(W). Hence it follows from [W, Prop.5.4.(d)], that p-linear groups are p-central p-groups with ΩEP. One can speculate whether every finite p-central p-group with ΩEP is indeed p-linear. This turns out to be the case for powerful p-central p-groups with ΩEPprovided p ≥ 5 (cf. [W,Th.7.1.]).

§5. F -isomorphisms in cohomology.

Let K be a perfect field of characteristic p and ϕ: A → B a morphism of graded commutative K-algebras. One says that ϕ is an F -isomorphism, if ker(ϕ) is nilpotent and im(ϕ^{▲}) contains the image of B_{red} under the n^{th}-Frobeniusmorphism

F^{n}(B_{red}): = { b^{p^{n}} | b ∈ B_{red} }  (5.1.)

for some n ∈ ℕ_{0}, where ϕ^{▲}: A_{red} → B_{red} denotes the induced morphism (cf. (3.8.)). Note that if A is a reduced K-algebra, then an F -isomorphism is injective.

In [Qu_{2}] it was shown, that for p odd, p-nilpotency can be characterized by the following: Let G be a finite group. Then G is p-nilpotent if and only if for P ∈ Syl_{p}(G),

res_{G,P}: H^{•}(G) ⟶ H^{•}(P)

is an F -isomorphism.

The main subject of this section is to investigate finite groups G which contain an elementary abelian p-subgroup A, such that

res_{G,A}: H^{•}(G) ⟶ H^{•}(A)

is an F -isomorphism. The basic tool for this treatment is Quillen’s stratification theorem (cf. [Qu_{1}]), which determines H^{•}(G, 𝔽̄_{p}) up to F -isomorphism, and his work on p-nilpotency (cf. [Qu_{2}]). We first recall some well-known facts about the reduced mod p cohomology ring H^{•}(G)_{red}.

Proposition 5.1. Let G be a finite group such that H^{•}(G)_{red} is a domain. Then G contains a unique G-conjugacy class of maximal elementary abelian p-subgroups of G.

Proof. (cf. [Qu_{2}, Cor. 3.2.])

For analyzing the structure of G, it is necessary to study the action of finite groups on polynomial algebras in characteristic p.

Lemma 5.2. Let K be a perfect field of characteristic p and let

A: = K[y_{1}, .., y_{n}] = ∐_{k∈ℕ_{0}} A_{k}

denote a polynomial graded K-algebra with |y_{i}| > 0. Let H be a finite group acting faithfully on A by homogeneous K-algebra automorphisms, i.e., h(A_{k}) ≤ A_{k} for all h ∈ H, k ∈ ℕ_{0}. Let

A^{H}: = { a ∈ A | h(a) = a, ∀h ∈ H }

denote the K-algebra of H-invariant elements of A. If the canonical embedding

ι: A^{H} ⟶ A

is an F -isomorphism, then H = 1.

Proof. F^{k}: A → A is an injective 𝔽_{p}-algebra morphism. Furthermore, since H acts by K-algebra automorphisms, one has

h ∘ F^{k} = F^{k} ∘ h  (5.2.)

for all k ∈ ℕ_{0}. By hypothesis there exists some n ∈ ℕ_{0} such that F^{n}(A) ≤ A^{H}. Then (5.2.) implies

F^{n}(a) = F^{n}(h(a))

for all a ∈ A, h ∈ H. Hence, the injectivity of F^{n} yields a = h(a) for all a ∈ A, h ∈ H, and the desired result follows.

As an immediate consequence of Lemma 5.2. and Proposition 5.1. one obtains:

Proposition 5.3. Let G be a finite group and let A ≤ G be an elementary abelian p-subgroup of G, such that

res_{G,A}: H^{•}(G) ⟶ H^{•}(A)

is an F -isomorphism. Then A is a maximal elementary abelian p-subgroup of G, N_{G}(A) = C_{G}(A), and { A^{g} | g ∈ G } is the unique G-conjugacy class of maximal elementary abelian p-subgroups of G.

Proof. Obviously, H^{•}(G)_{red} and H^{•}(A) must have the same Krull dimension. Hence, A has to be a maximal elementary abelian p-subgroup (cf. [Qu_{2}, Cor.3.3.]), and since H^{•}(G)_{red} is a domain, it follows from Proposition 5.1. that there is a unique G-conjugacy class of maximal elementary abelian p-subgroups of G.

Furthermore, the universal coefficient theorem implies that

res_{G,A}^{▲}: H^{•}(G, 𝔽̄_{p})_{red} ⟶ H^{•}(A, 𝔽̄_{p})_{red}

is also an F -isomorphism. Hence, one has (cf. [Ev, p.112f])

im(res_{G,A}^{▲}) ≤ (H^{•}(A, 𝔽̄_{p})_{red})^{W} ≤ H^{•}(A, 𝔽̄_{p})_{red},

where W: = N_{G}(A)/C_{G}(A), and W is acting faithfully on H^{•}(A, 𝔽̄_{p})_{red}. Thus,

ι: (H^{•}(A, 𝔽̄_{p})_{red})^{W} ⟶ H^{•}(A, 𝔽̄_{p})_{red}

is an F -isomorphism, and the desired result follows from Lemma 5.2.

For finite p-groups one has:

Proposition 5.4. Let G be a finite p-group, and assume that A ≤ G is a maximal elementary abelian p-subgroup of G such that N_{G}(A) = C_{G}(A). Then N_{G}(A) = C_{G}(A) = G, i.e., G is p-central and A = Ω_{1}(G) = Ω_{1}(Z(G)).

Proof. Assume the assertion is false and let G denote a minimal counterexample (with respect to |G|), i.e., N_{G}(A) ≠ G. Let M ⪇ G denote a maximal subgroup of G containing N_{G}(A). Then M is normal in G, A is a maximal elementary abelian p-subgroup of M and

N_{M}(A) = M ∩ N_{G}(A) = M ∩ C_{G}(A) = C_{M}(A).

Hence the assertion holds for M, i.e., A = Ω_{1}(Z(M)). In particular, A is characteristic in M and thus normal in G, a contradiction.

This yields the following:

Theorem 5.5. Let G be a finite group and A ≤ G an elementary abelian psubgroup of G such that

res_{G,A}: H^{•}(G) ⟶ H^{•}(A)

is an F -isomorphism. Then (a) for p ≠ 2, G is p-nilpotent, and P ∈ Syl_{p}(G) is a p-central group,

(b) for p = 2, P ∈ Syl_{2}(G) is a 2-central group.

Proof. By Proposition 5.3., A has to be a maximal elementary abelian psubgroup, and { A^{g} | g ∈ G } is the unique G-conjugacy class of maximal elementary abelian p-subgroups of G. Furthermore, N_{G}(A) = C_{G}(A). Let P ∈ Syl_{p}(G) with A ≤ P. Then

N_{P} (A) = P ∩ N_{G}(A) = P ∩ C_{G}(A) = C_{P} (A).

By Proposition 5.4., P must be p-central. Thus

res_{P,A}: H^{•}(P) ⟶ H^{•}(A)

is an F -isomorphism (cf. Prop.3.6.). Consider the commutative diagram

H^{•}(G) ⟶^{res_{G,P}} H^{•}(P)
∥  ↓^{res_{P,A}}
H^{•}(G) ⟶_{res_{G,A}} H^{•}(A)

By hypothesis and the previous remark, res_{G,A} and res_{P;A} are F -isomorphisms. Furthermore, res_{G,P} is injective, and thus has to be an F -isomorphism. Hence, G has to be p-nilpotent (cf. [Qu_{2}, Th.1.4.]).

One can use Theorem 5.5. to obtain the following well-known result (cf. [Hu, Kap.IV, Th.5.5]):

Corollary 5.6. Let G be a p-central group, p odd. Then G is p-nilpotent.

Proof. From Proposition 3.6. one concludes that

res_{G,Ω_{1}(G)}: H^{•}(G) ⟶ H^{•}(Ω_{1}(G))

is an F -isomorphism. Hence the desired result follows from Theorem 5.5.

Remark 5.7. For p = 2 a finite group as claimed in Theorem 5.5. does not have to be 2-nilpotent, e.g., G = C_{3} ⋉ Q_{8}, where C_{3} denotes the cyclic group of order 3 and Q_{8} the quaternion group of order 8, is not 2-nilpotent, but

res_{G,A}: H^{•}(G) ⟶ H^{•}(A),

A: = Ω_{1}(Q_{8}), is an F -isomorphism (cf. [Qu_{2}, Ex. after Th.1.4.]).

Theorem 5.8. Let G be a finite group and p be odd. Then

H^{•}(G)_{red} ≃ 𝔽_{p}[y_{1}, .., y_{n}],  (5.3.)

where all generators y_{i} have degree 2, if and only if G is p-nilpotent, P ∈ Syl_{p}(G) is a p-central group with ΩEP, and l-rk(P) = n.

Proof. If G is p-nilpotent, one has H^{•}(G) = H^{•}(P). Since P is p-central with ΩEP, one has

H^{•}(P)_{red} ≃ H^{•}(Ω_{1}(P))_{red}

(cf. Prop.3.6., Theorem 4.2.) and the claim follows in this case.

Assume that G is a group such that (5.3.) holds. Then G has a unique Gconjugacy class of maximal elementary abelian p-subgroups { A^{g} | g ∈ G } (cf. Prop. 5.1.). The restriction mapping

res_{G,A}^{▲}: H^{•}(G)_{red} ⟶ H^{•}(A)_{red}

§6. SOME SPECTRAL SEQUENCES OF p-CENTRAL EXTENSIONS. 21

must be injective (cf. [Qu_{2}, Th.3.1.]), and since H^{•}(G) and H^{•}(A) have the same Krull dimension (cf. [Qu_{2}, Cor.3.3.]) also bijective. Thus the restriction mapping res_{G,A}: H^{•}(G) → H^{•}(A) is an F -isomorphism and Theorem 5.5. applies, i.e., G is p-nilpotent and P ∈ Syl_{p}(G) is p-central. As H^{•}(G) ≃ H^{•}(P), Theorem 4.2. implies that P has the ΩEP and the theorem is proved.

§6. Some spectral sequences of p-central extensions.

Let G be a finite p-central p-group, p odd. In this section we investigate the Lyndon-Hochschild-Serre spectral sequence (LHS spectral sequence) associated to the extension

1 ⟶ A ⟶ G ⟶ Ḡ ⟶ 1  (†)

where A: = Ω_{1}(G) and Ḡ: = G/A. We restrict our considerations to the case where p is odd. Hence one knows by [W, Prop.5.4.(a)], that Ḡ is again a p-central group and Ḡ has the ΩEP. Thus

H^{•}(Ḡ) ≃ C ⊗ 𝔽_{p}[y_{1}, .., y_{n}],  (6.1.)

n = l-rk(Ḡ): = dim_{𝔽_{p}}(Ω_{1}(Ḡ)), |y_{i}| = 2, and C is a finite dimensional commutative 𝔽_{p}-algebra. (cf. Corollary 3.5., Theorem 4.2.). Since A is central in G, the LHS spectral sequence reduces to

E_{2}^{r,s} = H^{r}(Ḡ) ⊗ H^{s}(A) ⟹ H^{r+s}(G).  (6.2.)

Using the explicit description of the differential d_{2}: E_{2} → E_{2} given by L.S.Charlap and A.T.Vasquez (cf. [CV]), we obtain the following:

Lemma 6.1. Let G be a finite p-central p-group, p odd. Assume further that l-rk(Ḡ) = l-rk(G) = n. Then

E_{3}^{r,s} = C_{r} ⊗ S_{s},

where S_{s} denotes the s^{th}-homogeneous component of the 𝔽_{p}-algebra 𝔽_{p}[z_{1}, .., z_{n}], |z_{i}| = 2, and C = ∐_{r∈ℕ_{0}} C_{r} is given as in (6.1). Furthermore, the product structure on E_{3} is the naturally given one.

Proof. First we have to translate the description of d_{2} given in [CV] for our needs. Note that as (†) is a central extension, all the characteristc classes v^{N} vanish (cf. [CV,§7]). Hence, the diffential d_{2}: E_{2} → E_{2} depends only on the element α ∈ H^{2}(Ḡ, A) representing the extension (†) (cf. [CV, Main Theorem]). Thus it suffices to calculate

P_{∗}(α) ∈ H^{2}(Ḡ, 𝔽_{p}) ⊗ Hom_{𝔽_{p}}(H^{s}(A), H^{s−1}(A)).  (6.3.)

Let Δ: H^{•}(A) → H^{•}(A) ⊗ H^{•}(A) denote the coproduct induced by multiplication in A (cf. §3.). Hence, for

H^{•}(A) = Λ[a_{1}, .., a_{n}] ⊗ 𝔽_{p}[z_{1}, .., z_{n}]  (6.4)

one has

\begin{aligned}Δ(a_{i}) & = 1 ⊗ a_{i} + a_{i} ⊗ 1 \\ and  Δ(z_{j}) & = 1 ⊗ z_{j} + z_{j} ⊗ 1, \end{aligned}  (6.5)

and since Δ is a morphism of graded commutative 𝔽_{p}-algebras, this determines the map Δ completely. Let π_{1} ⊗ π_{s−1} denote the canonical projection

π_{1} ⊗ π_{s−1}: H^{•}(A) ⊗ H^{•}(A) ⟶ H^{1}(A) ⊗ H^{s−1}(A).

Then

τ_{s}: = (π_{1} ⊗ π_{s−1}) ∘ Δ: H^{s}(A) ⟶ H^{1}(A) ⊗ H^{s−1}(A)

has the form

τ_{q}(x) = ∑_{i=1}^{n} a_{i} ⊗ d_{i}(x),

where d_{i}: H^{•}(A) → H^{•}(A) are 𝔽_{p}-linear maps with the following properties:

(a) d_{i} are derivations of degree (−1)), i.e., for y, z ∈ H^{•}(A) homogeneous, one

has

d_{i}(y.z) = d_{i}(y).z + (−1)^{|y|}y.d_{i}(z),

(b)

d_{i}(a_{j}) = \{ \begin{matrix}1 & for i = j, \\ 0 & for i ≠ j, \end{matrix} , and d_{i}(z_{j}) = 0 for all i, j,

(c) d_{i} ∘ d_{j} = −d_{j} ∘ d_{i} for all i, j. Let α ∈ H^{2}(Ḡ, A) = H^{2}(Ḡ, 𝔽_{p}) ⊗ H_{1}(A, 𝔽_{p}) represent the extension (†). Since (†) is an Ω_{1}-extension, we may assume that

α = ∑_{i=1}^{n} y_{i} ⊗ a_{i}^{∗},  (6.6.)

where { a_{i}^{∗} | i = 1, .., n } ⊂ H_{1}(A, 𝔽_{p}) is the basis dual to { a_{i} | i = 1, .., n } ⊂ H^{1}(A, 𝔽_{p}). The element P_{∗}(α) is given by

P_{∗}(α) = ∑_{i=1}^{n} y_{i} ⊗ d_{i}  (6.7.)

(cf. [CV, §5]). For u ⊗ v ∈ E_{2}^{r,s} the image under d_{2} is given by

d_{2}(u ⊗ v) = − ∑_{i=1}^{n}(−1)^{r}.u.y_{i} ⊗ d_{i}(v).  (6.8.)

Hence, the subalgebra C ⊗ 𝔽_{p} ⊗ 𝔽_{p} ⊗ 𝔽_{p}[z_{1}, .., z_{n}] ≤ E_{2} is contained in the kernel of d_{2}. On the subalgebra

Γ: = 𝔽_{p} ⊗ 𝔽_{p}[y_{1}, .., y_{n}] ⊗ Λ[a_{1}, .., a_{n}] ⊗ 𝔽_{p}

the differential d_{2} induces a differential δ, such that

... ⟶^{δ} 𝔽_{p}[y_{1}, .., y_{n}] ⊗ Λ_{1}[a_{1}, .., a_{n}] ⟶^{δ} 𝔽_{p}[y_{1}, .., y_{n}] ⟶^{ε} 𝔽_{p}

is the Koszul-complex of the polynomial algebra 𝔽_{p}[y_{1}, .., y_{n}] (cf. [HS]), and thus H_{•}(Γ, δ) = 𝔽_{p} is concentrad in degree 0. Hence, E_{3} = H_{•}(E_{2}, d_{2}) ≃ C⊗𝔽_{p}[z_{1}, .., z_{n}], and the algebra structure on E_{3} is the natural algebra structure on C ⊗𝔽_{p}[z_{1}, .., z_{n}].

For k ≥ 2, the k^{th}-pages E_{k}^{•,•} are bigraded 𝔽_{p}-algebra, and the differentials d_{k}: E_{k}^{•,•} → E_{k}^{•,•} are derivations of total degree (−1) (cf. [Be]). Hence, the easy structure of the E_{3}-page makes it possible to calculate the cohomology ring H^{•}(G) in the following particular case:

Theorem 6.2. Let G be a finite p-central p-group with ΩEP, p odd. Assume further, that l-rk(Ḡ) = l-rk(G) = n. Let (E_{k}, d_{k}) denote the HLS spectral sequence associated to the extension (†). Then one has

E_{3}^{•,•} = E_{∞}^{•,•}  (6.9.)

and the product structure on E_{∞}^{•,•} coincides with the cup product on H^{•}(G). Furthermore, there exists an isomorphism

H^{•}(G) ≃ im(inf_{Ḡ,G}) ⊗ im(res_{G,A}).

Proof. By Lemma 6.1, we know that

E_{3}^{r,s} = C_{r} ⊗ S_{s},

where C = ∐_{r∈ℕ_{0}} C_{r} is as in (6.1), and S_{s} denotes the s^{th}-homogeneous part of the polynomial algebra 𝔽_{p}[z_{1}, .., z_{n}], |z_{i}| = 2 for all i. Since

d_{k}: E_{k}^{r,s} ⟶ E_{k}^{r+k,s−k+1},

it follows, that E_{k}^{r,0} is contained in the kernel of d_{k} for all k ≥ 2. Since G has the ΩEP, it follows that

dim_{𝔽_{p}}(im(res_{G,A}: H^{2}(G) → H^{2}(A))) ≥ n.  (6.10.)

Furthermore, one has a vertical edge homomorphism

e′: H^{2}(G) ⟶^{j} E_{∞}^{0,2}... ⟶^{f_{4}} E_{3}^{0,2} ⟶^{f_{3}} E_{2}^{0,2},

which coincides with res_{G,A}. Note that j is surjective, while f_{i}, i ≥ 3, are injective. Since dim_{𝔽_{p}}(E_{3}^{0,2}) = n, it follows from (6.10.) that all maps f_{i}, i ≥ 4, have to be bijective. Hence E_{k}^{0,2} ≤ ker(d_{k}) for all k ≥ 3. Since d_{k} is a derivation of E_{k}^{•,•} with respect to total degree, one follows easily by induction, that d_{k} ≡ 0 for all k ≥ 3. This yields (6.9.). By the same argument using the horizontal edge homomorphism one gets im(inf_{Ḡ,G}) ≃ E_{3}^{0,•}. Hence the desired result follows from Corollary 3.6. and the fact that d_{k}, k ≥ 2, are derivations.

Corollary 6.3. Let G be a powerful p-central p-group with ΩEP, p odd, and put n: = l-rk(G). Then

H^{•}(G) ≃ Λ[x_{1}, .., x_{n}] ⊗ 𝔽_{p}[y_{1}, .., y_{n}],

where |x_{i}| = 1, |y_{j}| = 2.

Proof. We procceed by induction on |G|. For G abelian the assertion is obvious. Hence assume that the assertion holds for all powerful p-central p-groups with ΩEP, which have order smaller than G.

If Ω_{1}(G) ≰ G^{p}, we choose a complement B for Ω_{1}(G) ∩ G^{p} in Ω_{1}(G). Hence

B ≃ B^{∗} = Ω_{1}(G).G^{p}/G^{p}.

We choose a complement H^{∗} to B^{∗} in G/G^{p}, and let H ≤ G denote its preimage under the canonical projection π: G → G/G^{p}. By construction H^{p} = G^{p} and hence, H is powerful p-central with ΩEP(cf. [W, Prop.5.4.(d)]). Furthermore, G = H ×B. Thus the assertion follows in this case.

Assume that Ω_{1}(G) ≤ G^{p}. Hence, l-rk(G) = l-rk(Ḡ), where Ḡ: = G/Ω_{1}(G). By [W, Prop.5.4.(a)], Ḡ is powerful p-central with ΩEP. Hence the desired result follows by induction and Theorem 6.2.

Next we consider another type of extension and the associated LHS spectral sequence. Let G be p-central with ΩEP and A ≤ Ω_{1}(G), |A| = p such that A ∩ Ω_{2}(G)^{p} = 1. Hence by Proposition 4.5. one knows that Ḡ: = G/A is also p-central with ΩEP. Let n = l-rk(G) = l-rk(Ḡ) + 1. Hence,

H^{•}(Ḡ) = D ⊗ 𝔽_{p}[y_{1}, .., y_{n−1}], H^{•}(A) = Λ[x] ⊗ 𝔽_{p}[y].  (6.11.)

We consider the extension

1 ⟶ A ⟶ G ⟶ Ḡ ⟶ 1  (‡)

and the associated LHS spectral sequence. Thus one has E_{2}^{•,•} = H^{•}(Ḡ) ⊗ H^{•}(A). As in the previous case we may assume that the cocycle

γ ∈ H^{2}(Ḡ, A) = H^{2}(Ḡ) ⊗ H_{1}(A)

representing the extension (‡) is given by γ = c ⊗ x^{∗} for some element c ∈ D_{2}. The element P_{∗}(γ) is then given by

P_{∗}(γ) = c ⊗ d,  (6.12.)

where d: H^{•}(A) → H^{•}(A) denotes the derivation given by d(x) = 1, d(y) = 0. Hence, for u ⊗ v ∈ E_{2}^{r,s} the image under d_{2} is given by

d_{2}(u ⊗ v) = −(−1)^{r}.u.c ⊗ d(v).  (6.13.)

Since 𝔽_{p} ⊗𝔽_{p}[y_{1}, .., y_{n−1}]⊗𝔽_{p} ⊗𝔽_{p}[y] is contained in the kernel of d_{2}, if follows that

E_{3} = 𝔽_{p}[y_{1}, .., y_{n−1}] ⊗ H_{•}(D ⊗ Λ[x], ∂) ⊗ 𝔽_{p}[y],

where ∂ denotes the induced differential on the algebra D ⊗ Λ[x]. More precisely we obtain:

Theorem 6.4. Let G be a p-central p-group, p odd, with ΩEP and A ≤ Ω_{1}(G), |A| = p, such that A ∩ Ω_{2}(G)^{p} = 1. Let n: = l-rk(G) and Ḡ: = G/A. Then H^{•}(Ḡ) = D ⊗ T, where T is a polynomial 𝔽_{p}-algebra of Krull dimension n − 1 generated in degree 2, and D is a finite-dimensional commutative 𝔽_{p}-algebra (cf. Cor. 3.5.). Let c ∈ D_{2} be given as in (6.12.), and let C^{•,•} denote the bigraded 𝔽_{p}-algebra

C^{•,0} = D ⊗ 1/D.c ⊗ 1, C^{•,1} = Ann_{D}(c) ⊗ x, C^{•,k} = 0 for k ≥ 2, (6.14.)

with the natural 𝔽_{p}-algebra structure. Then

E_{3}^{r,s} = ∑_{\begin{matrix}i+j=r \\ l+m=s \end{matrix}} T_{i} ⊗ C^{j,l} ⊗ S_{m},

where S = ∐_{m∈ℕ_{0}} S_{m} = 𝔽_{p}[y]. Furthermore, E_{3}^{•,•} = E_{∞}^{•,•}.

If D satisfies Poincaré duality in dimension n−1, the algebra Tot(C^{•,•}) satisfies Poincaré duality in dimension n.

Proof. By the previous remarks, we can identify C^{•,•} = H_{•}(D ⊗ Λ[x], ∂) and the differential ∂ is of bidegree (0, −1). This shows that

ker(∂) = D ⊗ 1 + Ann_{D}(c) ⊗ x
and  im(∂) = D.c ⊗ 1.

This yields (6.14.). If D satisfies Poincaré duality in dimension n − 1, it is easily seen that the product on Tot(C^{•,•}) induces a non-degenerated pairing between D ⊗ 1/D.c ⊗ 1 and Ann_{D}(c) ⊗ x. Hence, Tot(C^{•,•}) satisfies Poincaré duality in dimension n.

Let d_{k}: E_{k}^{r,s} → E_{k}^{r+k,s−k+1}, k ≥ 3, denote the higher differentials. Then one concludes as in the proof of Proposition 4.5. that T ⊗ 1 ⊗ 1, 1 ⊗ 1 ⊗ S ≤ ker(d_{k}). Since C^{•,•} is a bigraded algebra concentrated on the 0^{th} and 1^{st} buttom row, one has also C^{•,•} ≤ ker(d_{k}), k ≥ 3. This shows that E_{3}^{•,•} = E_{∞}^{•,•}.

Note that the product on H^{•}(G) does not necessarily coincide with the product on Tot(E_{3}^{∞}) = Tot(E_{∞}^{•,•}). However, if H^{•}(G) = R ⊗ 𝔽_{p}[y_{1}, .., y_{n}] one has R ≃ Tot(C^{•,•}) as 𝔽_{p}-modules. With the natural identification one gets for a ∈ C^{r,s}, b ∈ C^{u,v}

a ∘ b ≡ a.b + ⨁_{\begin{matrix}i+j=r+s+u+v \\ j≤s+v−1 \end{matrix}} C^{i,j},  (6.15.)

where “.” denotes the product in Tot(C^{•,•}) and “∘” denotes the product in R ≤ H^{•}(G) (cf. [B_{2}, Th. 3.9.3]). This yields:

Corollary 6.5. Let G and Ḡ be as in Theorem 6.4., H^{•}(G) = C ⊗𝔽_{p}[y_{1}, .., y_{n}], H^{•}(Ḡ) = D ⊗ 𝔽_{p}[y_{1}, .., y_{n−1}]. If D satisfies Poincaré duality in dimension n − 1, then C satisfies Poincaré duality in dimension n.

Proof. This follows from Theorem 6.4. and (6.15.).

§7.Conclusions and remarks.

Proof of Theorem B. The implication (1) ⇒ (2) is obvious. Furthermore, Theorem 5.8 implies (1) ⇒ (3). Hence, it remains to show that (3) implies (2). Without loss of generality we may assume that G is a p-central p-group with ΩEP. We proceed by induction on |G|. If G is elementary abelian there is nothing to prove. Thus we may assume that Ω_{1}(G) ⪇ G. The group G/Ω_{1}(G) is again pcentral with ΩEP(cf. [W, Prop.5.4.(a)]). If l-rk(G) = l-rk(G/Ω_{1}(G)), the claim follows by induction, Theorem 6.2. and Lemma 6.1. Hence we may assume that

n: = l-rk(G) > l-rk(G/Ω_{1}(G)).

This implies that Ω_{2}(G)^{p} ⪇ Ω_{1}(G) (cf. [W, Prop.4.1.(d)]). Let A ≤ Ω_{1}(G), |A| = p, such that A ∩ Ω_{2}(G)^{p} = 1. Then Ḡ: = G/A is again a p-central group with ΩEP (cf. Prop.4.5.). By induction, H^{•}(Ḡ) = D ⊗ 𝔽_{p}[y_{1}, .., y_{n−1}], where D is a finite dimensional 𝔽_{p}-algebra satisfying Poincaré duality in dimension n − 1. Then Corollary 3.6., Theorem 6.4. and Corollary 6.5. imply that H^{•}(G) = C ⊗ 𝔽_{p}[y_{1}, .., y_{n}], |y_{i}| = 2, and C is a finite-dimensional 𝔽_{p}-algebra satisfying Poincaré duality in dimension n.

Remark 7.1. (a) It might be possible to prove the implication (3) ⇒ (2) also by using the May spectral sequence. This would be the analogue of the treatment used in [La].

(b) The Poincaré duality of C can be shown also by the spectral sequence introduced by D.Benson and J.Carlson (cf. [BC, Th.5.5.]).

For M a finite-dimensional ℕ_{0}-graded 𝔽_{p}-module let

χ_{M}: = p_{M}(−1) = ∑_{k∈ℕ_{0}}(−1)^{k}. dim_{𝔽_{p}}(M_{k})  (7.1.)

denote the Euler characteristic. The proof of Theorem B can be also used to show the following property:

Proposition 7.2. Let G be a finite p-central p-group with ΩEP, p odd, and n = l-rk(G). Then H^{•}(G) = C ⊗ 𝔽_{p}[y_{1}, .., y_{n}], |y_{i}| = 2, C satisfies Poincaré duality in dimension n, and χ_{C} = 0.

Proof. If n is odd, Poincaré duality implies that χ_{C} = 0 (cf. (1.2.)). The general case can be proved by induction using Theorem 6.2. and 6.4. Note that with the same hypothesis as in Theorem 6.4. one has

p_{Tot(C^{•,•})}(t) = (1 − t^{2}).p_{D}(t) + (t + t^{2}).p_{Ann_{D}(c)}(t).

Proof of Corollary C. Corollary 6.3. yields that (2) ⇒ (1). On the other hand if G is a finite group such that H^{•}(G) = Λ[x_{1}, ..x_{n}] ⊗ 𝔽_{p}[y_{1}, .., y_{n}], |x_{i}| = 1, |y_{j}| = 2, p odd, then G is p-nilpotent, P ∈ Syl_{p}(G) is p-central with ΩEP and n = l-rk(P) (cf. Theorem 5.8.). Furthermore, one has also

P/[P, P].P^{p} = H_{1}(P, 𝔽_{p}) = 𝔽_{p}^{n}.

§7.CONCLUSIONS AND REMARKS. 27

But this implies that P is in addition powerful (cf. [W, Prop.4.3.]) and this completes the proof of the corollary.

Remark 7.3. Let A_{p} denote the Steenrod algebra for the prime p and A_{p}^{∗} ≤ A_{p} the subalgebra generated by the reduced power actions. Then it can be seen easily that for every powerful p-central group G with ΩEP, l-rk(G) = n, p odd, one has

H^{•}(G) ≃ H^{•}(𝔽_{p}^{n})

as A_{p}^{∗}-module. However, as it was pointed out to me by C.Broto and R.Levi, this isomorphism is not necessarily an isomorphism in the category of A_{p}-modules. It would be interesting to know whether the isomorphism type of G is determined by the Bockstein spectral sequence (cf. [BrL]).

References

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